yourstudent_geminifandomcom-20200216-history
Trigonometry
Trigonometry (from Greek trigōnon, "triangle" + metron, "measure" ) is a branch of mathematics that studies relationships involving lengths and angles of triangles. The field emerged during the 3rd century BC from applications of geometry to astronomical studies.R. Nagel (ed.), Encyclopedia of Science, 2nd Ed., The Gale Group (2002) The 3rd-century astronomers first noted that the lengths of the sides of a right-angle triangle and the angles between those sides have fixed relationships: that is, if at least the length of one side and the value of one angle is known, then all other angles and lengths can be determined algorithmically. These calculations soon came to be defined as the trigonometric functions and today are pervasive in both pure and applied mathematics: fundamental methods of analysis such as the Fourier transform, for example, or the wave equation, use trigonometric functions to understand cyclical phenomena across many applications in fields as diverse as physics, mechanical and electrical engineering, music and acoustics, astronomy, ecology, and biology. Trigonometry is also the foundation of the practical art of surveying. Trigonometry is most simply associated with planar right-angle triangles (each of which is a two-dimensional triangle with one angle equal to 90 degrees). The applicability to non-right-angle triangles exists, but, since any non-right-angle triangle (on a flat plane) can be bisected to create two right-angle triangles, most problems can be reduced to calculations on right-angle triangles. Thus the majority of applications relate to right-angle triangles. One exception to this is spherical trigonometry, the study of triangles on spheres, surfaces of constant positive curvature, in elliptic geometry (a fundamental part of astronomy and navigation). Trigonometry on surfaces of negative curvature is part of hyperbolic geometry. Trigonometry basics are often taught in school, either as a separate course or as a part of a precalculus course. History was apparently compiled by Hipparchus, who is now consequently known as "the father of trigonometry." ]] Sumerian astronomers studied angle measure, using a division of circles into 360 degrees.Aaboe, Asger. Episodes from the Early History of Astronomy. New York: Springer, 2001. ISBN 0-387-95136-9 They, and later the Babylonians, studied the ratios of the sides of similar triangles and discovered some properties of these ratios, but did not turn that into a systematic method for finding sides and angles of triangles. The ancient Nubians used a similar method. The ancient Greeks transformed trigonometry into an ordered science."The Beginnings of Trigonometry". Rutgers, The State University of New Jersey. In the 3rd century BCE, classical Greek mathematicians (such as Euclid and Archimedes) studied the properties of chords and inscribed angles in circles, and proved theorems that are equivalent to modern trigonometric formulae, although they presented them geometrically rather than algebraically. Claudius Ptolemy expanded upon Hipparchus' Chords in a Circle in his Almagest.Marlow Anderson, Victor J. Katz, Robin J. Wilson (2004). Sherlock Holmes in Babylon: and other tales of mathematical history. MAA. p. 36. ISBN 0-88385-546-1 The modern sine function was first defined in the Surya Siddhanta, and its properties were further documented by the 5th century (CE) Indian mathematician and astronomer Aryabhata.Boyer p. 215 These Greek and Indian works were translated and expanded by medieval Islamic mathematicians. By the 10th century, Islamic mathematicians were using all six trigonometric functions, had tabulated their values, and were applying them to problems in spherical geometry. At about the same time, Chinese mathematicians developed trigonometry independently, although it was not a major field of study for them. Knowledge of trigonometric functions and methods reached Europe via Latin translations of the works of Persian and Arabic astronomers such as Al Battani and Nasir al-Din al-Tusi.Boyer pp. 237, 274 One of the earliest works on trigonometry by a European mathematician is De Triangulis by the 15th century German mathematician Regiomontanus. Trigonometry was still so little known in 16th-century Europe that Nicolaus Copernicus devoted two chapters of De revolutionibus orbium coelestium to explain its basic concepts. Driven by the demands of navigation and the growing need for accurate maps of large geographic areas, trigonometry grew into a major branch of mathematics. Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595. Gemma Frisius described for the first time the method of triangulation still used today in surveying. It was Leonhard Euler who fully incorporated complex numbers into trigonometry. The works of James Gregory in the 17th century and Colin Maclaurin in the 18th century were influential in the development of trigonometric series.William Bragg Ewald (2008). From Kant to Hilbert: a source book in the foundations of mathematics. Oxford University Press US. p. 93. ISBN 0-19-850535-3 Also in the 18th century, Brook Taylor defined the general Taylor series.Kelly Dempski (2002). Focus on Curves and Surfaces. p. 29. ISBN 1-59200-007-X Overview If one angle of a triangle is 90 degrees and one of the other angles is known, the third is thereby fixed, because the three angles of any triangle add up to 180 degrees. The two acute angles therefore add up to 90 degrees: they are complementary angles. The shape of a triangle is completely determined, except for similarity, by the angles. Once the angles are known, the ratios of the sides are determined, regardless of the overall size of the triangle. If the length of one of the sides is known, the other two are determined. These ratios are given by the following trigonometric functions of the known angle A'', where ''a, '' b'' and c'' refer to the lengths of the sides in the accompanying figure: *'Sine' function (sin), defined as the ratio of the side opposite the angle to the hypotenuse. :: \sin A=\frac{\textrm{opposite}}{\textrm{hypotenuse}}=\frac{a}{\,c\,}\,. *'Cosine' function (cos), defined as the ratio of the adjacent leg to the hypotenuse. :: \cos A=\frac{\textrm{adjacent}}{\textrm{hypotenuse}}=\frac{b}{\,c\,}\,. *'Tangent' function (tan), defined as the ratio of the opposite leg to the adjacent leg. :: \tan A=\frac{\textrm{opposite}}{\textrm{adjacent}}=\frac{a}{\,b\,}=\frac{\sin A}{\cos A}\,. The '''hypotenuse' is the side opposite to the 90 degree angle in a right triangle; it is the longest side of the triangle, and one of the two sides adjacent to angle A''. The '''adjacent leg' is the other side that is adjacent to angle A''. The '''opposite side' is the side that is opposite to angle A''. The terms '''perpendicular' and base are sometimes used for the opposite and adjacent sides respectively. Many people find it easy to remember what sides of the right triangle are equal to sine, cosine, or tangent, by memorizing the word SOH-CAH-TOA (see below under Mnemonics). The reciprocals of these functions are named the cosecant (csc or cosec), secant (sec), and cotangent (cot), respectively: : \csc A=\frac{1}{\sin A}=\frac{c}{a} , : \sec A=\frac{1}{\cos A}=\frac{c}{b} , : \cot A=\frac{1}{\tan A}=\frac{\cos A}{\sin A}=\frac{b}{a} . The inverse functions are called the arcsine, arccosine, and arctangent, respectively. There are arithmetic relations between these functions, which are known as trigonometric identities. The cosine, cotangent, and cosecant are so named because they are respectively the sine, tangent, and secant of the complementary angle abbreviated to "co-". With these functions one can answer virtually all questions about arbitrary triangles by using the law of sines and the law of cosines. These laws can be used to compute the remaining angles and sides of any triangle as soon as two sides and their included angle or two angles and a side or three sides are known. These laws are useful in all branches of geometry, since every polygon may be described as a finite combination of triangles. Extending the definitions The above definitions apply to angles between 0 and 90 degrees (0 and π/2 radians) only. Using the unit circle, one can extend them to all positive and negative arguments (see trigonometric function). The trigonometric functions are periodic, with a period of 360 degrees or 2π radians. That means their values repeat at those intervals. The tangent and cotangent functions also have a shorter period, of 180 degrees or π radians. The trigonometric functions can be defined in other ways besides the geometrical definitions above, using tools from calculus and infinite series. With these definitions the trigonometric functions can be defined for complex numbers. The complex exponential function is particularly useful. : e^{x+iy} = e^x(\cos y + i \sin y). See Euler's and De Moivre's formulas. Image:Sine curve drawing animation.gif|Graphing process of y'' = sin(''x) using a unit circle. Image:tan drawing process.gif|Graphing process of y'' = tan(''x) using a unit circle. Image:csc drawing process.gif|Graphing process of y'' = csc(''x) using a unit circle. Mnemonics A common use of mnemonics is to remember facts and relationships in trigonometry. For example, the sine, cosine, and tangent ratios in a right triangle can be remembered by representing them as strings of letters. For instance, a mnemonic is SOH-CAH-TOA: :S'ine = '''O'pposite ÷ 'H'ypotenuse :'C'osine = 'A'djacent ÷ 'H'ypotenuse :'T'angent = 'O'pposite ÷ 'A'djacent One way to remember the letters is to sound them out phonetically (i.e., SOH-CAH-TOA, which is pronounced 'so-kə-'''toe-uh' ). Another method is to expand the letters into a sentence, such as "S'ome '''O'ld 'H'ippy 'C'aught 'A'nother 'H'ippy 'T'rippin' 'O'n 'A'cid".A sentence more appropriate for high schools is "Some old horse came a'hopping through our alley". Calculating trigonometric functions Trigonometric functions were among the earliest uses for mathematical tables. Such tables were incorporated into mathematics textbooks and students were taught to look up values and how to interpolate between the values listed to get higher accuracy. Slide rules had special scales for trigonometric functions. Today scientific calculators have buttons for calculating the main trigonometric functions (sin, cos, tan, and sometimes cis and their inverses). Most allow a choice of angle measurement methods: degrees, radians, and sometimes grad. Most computer programming languages provide function libraries that include the trigonometric functions. The floating point unit hardware incorporated into the microprocessor chips used in most personal computers have built-in instructions for calculating trigonometric functions. Applications of trigonometry s are used to measure the angle of the sun or stars with respect to the horizon. Using trigonometry and a marine chronometer, the position of the ship can be determined from such measurements.]] There are an enormous number of uses of trigonometry and trigonometric functions. For instance, the technique of triangulation is used in astronomy to measure the distance to nearby stars, in geography to measure distances between landmarks, and in satellite navigation systems. The sine and cosine functions are fundamental to the theory of periodic functions such as those that describe sound and light waves. Fields that use trigonometry or trigonometric functions include astronomy (especially for locating apparent positions of celestial objects, in which spherical trigonometry is essential) and hence navigation (on the oceans, in aircraft, and in space), music theory, audio synthesis, acoustics, optics, analysis of financial markets, electronics, probability theory, statistics, biology, medical imaging (CAT scans and ultrasound), pharmacy, chemistry, number theory (and hence cryptology), seismology, meteorology, oceanography, many physical sciences, land surveying and geodesy, architecture, phonetics, economics, electrical engineering, mechanical engineering, civil engineering, computer graphics, cartography, crystallography and game development. Pythagorean identities Identities are those equations that hold true for any value. : \sin^2 A + \cos^2 A = 1 \ : \sec^2 A - \tan^2 A = 1 \ : \csc^2 A - \cot^2 A = 1 \ Angle transformation formulas : \sin (A \pm B) = \sin A \ \cos B \pm \cos A \ \sin B : \cos (A \pm B) = \cos A \ \cos B \mp \sin A \ \sin B : \tan (A \pm B) = \frac{ \tan A \pm \tan B }{ 1 \mp \tan A \ \tan B} : \cot (A \pm B) = \frac{ \cot A \ \cot B \mp 1}{ \cot B \pm \cot A } Common formulas Certain equations involving trigonometric functions are true for all angles and are known as trigonometric identities. Some identities equate an expression to a different expression involving the same angles. These are listed in List of trigonometric identities. Triangle identities that relate the sides and angles of a given triangle are listed below. In the following identities, A'', ''B and C'' are the angles of a triangle and ''a, b'' and ''c are the lengths of sides of the triangle opposite the respective angles. Law of sines The '''law of sines (also known as the "sine rule") for an arbitrary triangle states: : \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R, where R'' is the radius of the circumscribed circle of the triangle: : R = \frac{abc}{\sqrt{(a+b+c)(a-b+c)(a+b-c)(b+c-a)}}. Another law involving sines can be used to calculate the area of a triangle. Given two sides and the angle between the sides, the area of the triangle is: : \mbox{Area} = \frac{1}{2}a b\sin C. s of an angle ''θ can be constructed geometrically in terms of a unit circle centered at O''.]] Law of cosines The 'law of cosines' (known as the cosine formula, or the "cos rule") is an extension of the Pythagorean theorem to arbitrary triangles: : c^2=a^2+b^2-2ab\cos C ,\, or equivalently: : \cos C=\frac{a^2+b^2-c^2}{2ab}.\, Law of tangents The 'law of tangents': : \frac{a-b}{a+b}=\frac{\tan\left\tfrac{1}{2}(A-B)\right}{\tan\left\tfrac{1}{2}(A+B)\right} Euler's formula Euler's formula, which states that e^{ix} = \cos x + i \sin x , produces the following analytical identities for sine, cosine, and tangent in terms of ''e and the imaginary unit i: : \sin x = \frac{e^{ix} - e^{-ix}}{2i}, \qquad \cos x = \frac{e^{ix} + e^{-ix}}{2}, \qquad \tan x = \frac{i(e^{-ix} - e^{ix})}{e^{ix} + e^{-ix}}. See also * Aryabhata's sine table * Generalized trigonometry * Lénárt sphere * List of triangle topics * List of trigonometric identities * Rational trigonometry * Skinny triangle * Small-angle approximation * Trigonometric functions * Trigonometry in Galois fields * Unit circle * Uses of trigonometry References Bibliography * * *Christopher M. Linton (2004). From Eudoxus to Einstein: A History of Mathematical Astronomy . Cambridge University Press. *Weisstein, Eric W. "Trigonometric Addition Formulas". Wolfram MathWorld. Weiner. External links * Khan Academy: Trigonometry, free online micro lectures * Trigonometry by Alfred Monroe Kenyon and Louis Ingold, The Macmillan Company, 1914. In images, full text presented. * Benjamin Banneker's Trigonometry Puzzle at Convergence * Dave's Short Course in Trigonometry by David Joyce of Clark University *Trigonometry, by Michael Corral, Covers elementary trigonometry, Distributed under GNU Free Documentation License Category:Trigonometry Category:Greek inventions